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Given below are two statements:

Statement $I$: The laws of nature put two fundamental limits on data rate of a channel. The H.Nyquist limit restricts the number of independent samples per second to twice the band-width in a Noiseless channel

Statement $II$: Shannon’s major result about noised channel is that maximum data rate of a channel whose band width is $H$ Hz, and whose signal-to-noise ratio is $S/N$ is given by: Maximum number of bits/sec$=H \log _2 ( 1 + \dfrac{S}{N})$

In the light of the above statements, choose the correct answer from the options given below

  1. Both Statement $I$ and Statement $II$ are true
  2. Both Statement $I$ and Statement $II$ are false
  3. Statement $I$ is correct but Statement $II$ is false
  4. Statement $I$ is incorrect but Statement $II$ is true
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ANS is option A Both are true 

This is from  forouzan chapter 3 

One of the main objectives of data communications is to increase the data rate. There are three factors that determine the data rate of a channel:

  • Bandwidth of the channel
  • Number of levels of signals that are used
  • Noise present in the channel

Data rate can be calculated using two theoretical formulae:

  • a) Nyquist Bit Rate – for noiseless channel    b)Shannon’s Capacity – for noisy channel

 

  • Noiseless Channel: Nyquist Bit Rate
    For a noiseless channel, the Nyquist bit rate formula defines the theoretical maximum
    bit rate

    Bit Rate = 2 x bandwidth x lg L
    In this formula, bandwidth is the bandwidth of the channel, L is the number of signal
    levels used to represent data, and Bit Rate is the bit rate in bits per second
  • According to the formula, we might think that, given a specific bandwidth, we can
    have any bit rate we want by increasing the number of signa11eve1s. Although the idea
    is theoretically correct, practically there is a limit. When we increase the number of signal1eve1s,
    we impose a burden on the receiver. If the number of levels in a signal is just 2,
    the receiver can easily distinguish between a 0 and a 1. If the level of a signal is 64, the
    receiver must be very sophisticated to distinguish between 64 different levels. In other
    words, increasing the levels of a signal reduces the reliability of the system.                                                                                                                                                                                                          Shannon capacity, to determine the
  • theoretical highest data rate for a noisy channel:
    Capacity =bandwidth X lg2(1+SNR)
  • SNR is signal to noise ratio
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